100=X(X+1)

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Solution for 100=X(X+1) equation:


Simplifying
100 = X(X + 1)

Reorder the terms:
100 = X(1 + X)
100 = (1 * X + X * X)
100 = (1X + X2)

Solving
100 = 1X + X2

Solving for variable 'X'.

Reorder the terms:
100 + -1X + -1X2 = 1X + -1X + X2 + -1X2

Combine like terms: 1X + -1X = 0
100 + -1X + -1X2 = 0 + X2 + -1X2
100 + -1X + -1X2 = X2 + -1X2

Combine like terms: X2 + -1X2 = 0
100 + -1X + -1X2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-100 + X + X2 = 0

Move the constant term to the right:

Add '100' to each side of the equation.
-100 + X + 100 + X2 = 0 + 100

Reorder the terms:
-100 + 100 + X + X2 = 0 + 100

Combine like terms: -100 + 100 = 0
0 + X + X2 = 0 + 100
X + X2 = 0 + 100

Combine like terms: 0 + 100 = 100
X + X2 = 100

The X term is X.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + X2 = 100 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + X2 = 100 + 0.25

Combine like terms: 100 + 0.25 = 100.25
1.25 + X2 = 100.25

Factor a perfect square on the left side:
(X + 0.5)(X + 0.5) = 100.25

Calculate the square root of the right side: 10.012492197

Break this problem into two subproblems by setting 
(X + 0.5) equal to 10.012492197 and -10.012492197.

Subproblem 1

X + 0.5 = 10.012492197 Simplifying X + 0.5 = 10.012492197 Reorder the terms: 0.5 + X = 10.012492197 Solving 0.5 + X = 10.012492197 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + X = 10.012492197 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = 10.012492197 + -0.5 X = 10.012492197 + -0.5 Combine like terms: 10.012492197 + -0.5 = 9.512492197 X = 9.512492197 Simplifying X = 9.512492197

Subproblem 2

X + 0.5 = -10.012492197 Simplifying X + 0.5 = -10.012492197 Reorder the terms: 0.5 + X = -10.012492197 Solving 0.5 + X = -10.012492197 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + X = -10.012492197 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = -10.012492197 + -0.5 X = -10.012492197 + -0.5 Combine like terms: -10.012492197 + -0.5 = -10.512492197 X = -10.512492197 Simplifying X = -10.512492197

Solution

The solution to the problem is based on the solutions from the subproblems. X = {9.512492197, -10.512492197}

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